The document discusses how an isospin asymmetry affects the phase structure near a chiral tricritical point in quark matter. Using a Ginzburg-Landau analysis, it finds that the tricritical point shifts and splits into new critical points with increasing isospin density. This leads to large regions of the phase diagram being occupied by homogeneous and solitonic pion condensates instead of chiral condensates.

1. μI
Taking
the
perturbative
expansion
with
for
the
contribution
of
quark
loop,
resulting
in
deriving
the
relation
of
GL
couplings
where
refers
to
the
quantity
in
the
absence
of
and
is
taken
away
in
following
text
µI
Structure change of the chiral critical point driven by isospin density
Department of Physics, Tokyo University of Science, Tokyo 162-8601, Japan
QCD@Work @ Lecce - Italy, 18-21 June 2012
Yuhei Iwata, Hiroaki Abuki and Katsuhiko Suzuki
1. Introduction
Abstract
4. Summary
We
study
the
influence
of
the
isospin
asymmetry
in
strong
interacting
quark
matters
near
the
chiral
tricritical
point
(TCP).
At
the
vicinity
of
TCP
we
find
the
drastic
change
of
the
phase
diagram:
shifting
of
TCP
and
splitting
of
TCP
into
new
four
critical
points.
Accordingly
the
homogeneous
pion
condensate
and
solitonic
one
come
to
occupy
large
domains
in
the
phase
diagram.
Reference
We
showed
how
the
isospin
asymmetry
affects
the
phase
structure
near
the
TCP
and
found
the
remarkable
effects,
e.g.
a
shift
of
TCP
location
and
splitting
of
TCP.
Further,
we
found
the
homogeneous
and
solitonic
pion
condensates
have
large
region
instead
of
chiral
condensates
on
the
phase
diagram
of
GL
coupling
space.
[1]
R.
Casalbuoni,
PoS
CPOD
2006,
001
(2006).
[2]
D.
Nickel,
Phys.
Rev.
Lett.
103,
072301
(2009).
[3]
H.
Abuki,
D.
Ishibashi
and
K.
Suzuki,
Phys.
Rev.
D
85,
074002
(2012).
[4]
A.
I.
Buzdin
and
H.
Kachkachi,
Phys.
Lett.
A225,
341
(1997).
2. Method
3. Results
our objective
We
investigate
how
the
phase
structures
near
TCP
changes
at
finite
focusing
on
the
isospin
density
µI
In
the
result
of
variation
:
Using
GL
analysis
for
describing
phase
structures
around
TCP
solid
line
:
1st
order
dashed
line:
2nd
order
mapping
Phase
diagram
with
inhomogeneous
states
included
.Phase
diagram
with
homogeneous
states
only
.
fine
structures
appear
below
TCP’
solitonic
pion
condensate
dominate
the
fine
structures
at
,
a
major
part
of
solitonic
σ(x)
taken
over
by
π(x)
CEP
χSB
(σ)
Wigner
T
μq
1st
order
crossover
our
focus
μI
?µI = µu µd
mq = 0
QCD
phase
diagram
1st
order
crossover
2nd
order
in
GL
parameter
space
We
study
the
effect
of
a
finite
quark
mass
and
an
isospin
asymmetry
for
realistic
systems
on
the
chiral
tricritical
point
(TCP)[1]
The
quark
mass
turns
TCP
into
CEP
simply,
but
doesn’t
change
phases
The
isospin
asymmetry
which,
for
example,
arises
from
the
neutrality
affects
phase
structures
at
vacuum:
the
emergence
of
pion
condensate
focus
on
the
isospin
density
= ( , 1, 2, 3)order
parameter
:
chiral
four
vector
c
Assume
:
quark
loop
contribution
is
dominant
over
gluonic
one
Justified:
if
TCP
is
located
at
large
fugacity
region eµc/Tc
1
Ginzburg-Landau potential.
Perturbative expansion.
π
occupies
large
domain
in
the
phase
diagram
bicritical
point
P
appears
among
π,
σ,
Wigner
phases.
TCP
shifts
to
(0, µ2
I)
GL[ , 0] =
2
2
2
+
4 + µ2
I
4
4
+
1
6
6
.
0 0
TCP’
changes
into
a
Lifshitz
point
and
splits
into
three
critical
points
:
P,
Q,
R
4 µ2
I
-­‐term
favors
the
pion
condensate
π
with
(x) = k sn(kz; ) ,
µI
2
4 > 0
(0)
2
4
6
=
1 aµ2
I O(µ4
I)
0 1 bµ2
I
0 0 1
(0)
2
(0)
4
(0)
6
,
2
4 [or 4b(c)]
=
cµ2
I O(µ4
I)
0 dµ2
I [db(c)µ2
I]
(0)
4
(0)
6
.
SU(2)L SU(2)R
SU(2)L SU(2)R
U(1)I3;V U(1)I3;A
Chiral
symmetry
Explicit
breaking
of
symmetry
h
ignoring
quark
mass
term,
considering
in
future
GL[ (x), (x)] =
2
2
2
+
4
4
( 2
)2
+
4,b
4
( )2
+
6
6
( 2
)3
+
6,b
6
( , )2
+
6,c
6
[ 2
( )2
( , )2
] +
6,d
6
( )2
2
2
2
c
4
4
4
c
4,b
4
( 2 2
c ) 2
c
4,c
4
( 2
c )2
a = 0, b = 1, c = 1/2, d = db(c) = 1
For
chiral
symmetric
term[2] 4,b = 4 , ( 6,b, 6,c, 6,d) = (5, 3, 1/2) 6
arXiv:1206.2870
U(1)I3;V U(1)I3;A
U(1)I3;V
U(1)I3;A
solitonic solitonic
U(1)I3;V U(1)I3;A
U(1)I3;V
U(1)I3;A
[4]
Let
us
discuss
the
phase
diagram
in
-­‐
space
We
get
the
GL
potential
parametrized
by
and
scale
every
quantity
in
the
unit
,
having
[3]6 = 1[ 6]
{ 2, 4, 6, µ2
I}
( 2, 4, µ2
I)